FAQ: Should the sum in Bode's integral formula be over the real part of the unstable poles?

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(Contributed by Richard Murray, 5 Apr 09)

Bode's integral formula is given by


where the sum is over all of the unstable poles math. In the proof (at the end of Section 11.5), the sum is over the real parts of the unstable poles.

In the book we consider only transfer functions with real coefficients and so all poles come in complex conjugate pairs. Because of this, the imaginary parts cancel out in the sum and it should be OK without explicitly specifying the real parts of the poles.